$$a_n=\frac {(x+a_{n-1})^2}{1+a_{n-1}}$$
$x = 0.1$ converges to $0.0125$
$x = 0.2$ converges to $ 0.0666…$
$x = 0.3$ converges to $0.225$
$x = 0.4 $ converges to $ 0.8$
For $0 < x < 1$, is there mathematically a trend or pattern to where $a_n$ converges for different values of $x$, and if yes, an equation that can predict that?
Background and motivation, why is this important, I'm working on a game theoretical system. A collusion attack vector scales with $a_n$